Your computations are correct. The fundamental difficulty is that one cannot generally expect more than
a couple of places of accuracy from a normal approximation to a Poisson distribution.

For your problem, it may be best to look at the complementary probabilities in the right tail.

    > 1-ppois(687, 625)
    [1] 0.006821267
    > 1-pnorm(687.5, 625, 25)
    [1] 0.006209665
    > 1-pnorm(687, 625, 25)
    [1] 0.006569119

From close inspection of the plot below, one can see that the normal approximation already slightly
underestimates the right-tail probability. The continuity correction takes
away a little probability from that tail, which in this case happens to make
it even worse.

[![enter image description here][1]][1]


The continuity correction usually improves the approximation, but that may be true only when the approximation is _already very good._ In your problem the
approximation is not good enough for a discussion of the third and fourth decimal
places to be productive.

  [1]: https://i.sstatic.net/Im4G5.png